Properties

Label 21.14.g.a
Level $21$
Weight $14$
Character orbit 21.g
Analytic conductor $22.518$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 21.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.5184950799\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -729 - 729 \zeta_{6} ) q^{3} + ( -8192 + 8192 \zeta_{6} ) q^{4} + ( 52418 + 281733 \zeta_{6} ) q^{7} + 1594323 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -729 - 729 \zeta_{6} ) q^{3} + ( -8192 + 8192 \zeta_{6} ) q^{4} + ( 52418 + 281733 \zeta_{6} ) q^{7} + 1594323 \zeta_{6} q^{9} + ( 11943936 - 5971968 \zeta_{6} ) q^{12} + ( 235871 - 471742 \zeta_{6} ) q^{13} -67108864 \zeta_{6} q^{16} + ( -426993846 + 213496923 \zeta_{6} ) q^{19} + ( 167170635 - 448979436 \zeta_{6} ) q^{21} + ( 1220703125 - 1220703125 \zeta_{6} ) q^{25} + ( 1162261467 - 2324522934 \zeta_{6} ) q^{27} + ( -2737364992 + 429408256 \zeta_{6} ) q^{28} + ( -4748529235 - 4748529235 \zeta_{6} ) q^{31} -13060694016 q^{36} -27377427169 \zeta_{6} q^{37} + ( -515849877 + 515849877 \zeta_{6} ) q^{39} + 77218825315 q^{43} + ( -48922361856 + 97844723712 \zeta_{6} ) q^{48} + ( -76625836565 + 108909244077 \zeta_{6} ) q^{49} + ( 1932255232 + 1932255232 \zeta_{6} ) q^{52} + 466917770601 q^{57} + ( -282783926872 + 141391963436 \zeta_{6} ) q^{61} + ( -449173401759 + 532744624773 \zeta_{6} ) q^{63} + 549755813888 q^{64} + ( 226871917261 - 226871917261 \zeta_{6} ) q^{67} + ( -1044098651241 - 1044098651241 \zeta_{6} ) q^{73} + ( -1779785156250 + 889892578125 \zeta_{6} ) q^{75} + ( 1748966793216 - 3497933586432 \zeta_{6} ) q^{76} -4084540747547 \zeta_{6} q^{79} + ( -2541865828329 + 2541865828329 \zeta_{6} ) q^{81} + ( 2308577697792 + 1369461841920 \zeta_{6} ) q^{84} + ( 145269174964 - 91180416599 \zeta_{6} ) q^{91} + 10385033436945 \zeta_{6} q^{93} + ( 6234825725848 - 12469651451696 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2187q^{3} - 8192q^{4} + 386569q^{7} + 1594323q^{9} + O(q^{10}) \) \( 2q - 2187q^{3} - 8192q^{4} + 386569q^{7} + 1594323q^{9} + 17915904q^{12} - 67108864q^{16} - 640490769q^{19} - 114638166q^{21} + 1220703125q^{25} - 5045321728q^{28} - 14245587705q^{31} - 26121388032q^{36} - 27377427169q^{37} - 515849877q^{39} + 154437650630q^{43} - 44342429053q^{49} + 5796765696q^{52} + 933835541202q^{57} - 424175890308q^{61} - 365602178745q^{63} + 1099511627776q^{64} + 226871917261q^{67} - 3132295953723q^{73} - 2669677734375q^{75} - 4084540747547q^{79} - 2541865828329q^{81} + 5986617237504q^{84} + 199357933329q^{91} + 10385033436945q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1093.50 + 631.333i −4096.00 7094.48i 0 0 193284. 243988.i 0 797162. 1.38072e6i 0
17.1 0 −1093.50 631.333i −4096.00 + 7094.48i 0 0 193284. + 243988.i 0 797162. + 1.38072e6i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.14.g.a 2
3.b odd 2 1 CM 21.14.g.a 2
7.d odd 6 1 inner 21.14.g.a 2
21.g even 6 1 inner 21.14.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.14.g.a 2 1.a even 1 1 trivial
21.14.g.a 2 3.b odd 2 1 CM
21.14.g.a 2 7.d odd 6 1 inner
21.14.g.a 2 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{14}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1594323 + 2187 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 96889010407 - 386569 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 166905385923 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 136742808391403787 + 640490769 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 67645589686949055675 + 14245587705 T + T^{2} \)
$37$ \( \)\(74\!\cdots\!61\)\( + 27377427169 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -77218825315 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( \)\(59\!\cdots\!88\)\( + 424175890308 T + T^{2} \)
$67$ \( \)\(51\!\cdots\!21\)\( - 226871917261 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( \)\(32\!\cdots\!43\)\( + 3132295953723 T + T^{2} \)
$79$ \( \)\(16\!\cdots\!09\)\( + 4084540747547 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( \)\(11\!\cdots\!12\)\( + T^{2} \)
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